On Actions of Tori and Quaternionic Tori on Products of Spheres

We study the actions of tori (standard compact tori as well as their quaternionic analogs) on products of spheres. We prove that the orbit space of a specific action of a torus on a product of spheres is homeomorphic to a sphere. A similar statement for the real torus $\mathbb Z_2^n$ was proved by the second author in 2019. We also extend this result to arbitrary compact topological groups, thus generalizing the results mentioned above as well as the results of the first author on the actions of a compact torus of complexity $1$.